Numerical Algorithms. IE 496 Lecture 20

Transcription

1 Numerical Algorithms IE 496 Lecture 20

2 Reading for This Lecture Primary Miller and Boxer, Pages Forsythe and Mohler, Sections 1 and 2

3 Numerical Algorithms

4 Numerical Analysis So far, we have looked primarily at algorithms for discrete problems. Now we will consider problems from continuous mathematics. Numerical analysis is the study of algorithms for these problems. The main difference between the two areas is that in continuous mathematics, numbers must be approximated in general.

5 Problems and Algorithms A problem is a map from f: X Y, where X and Y are normed vector spaces. A numerical algorithm is a procedure which calculates F(x) Y, an approximation of f(x). A numerical algorithm does not necessatily have to be finite. Some algorithms converge (hopefully quickly) to the true solution "in the limit".

6 Conditioning A problem is well-conditioned if x x f(x ) f(x). Otherwise, it is ill-conditioned. Notice that well-conditioned requires all small perturbations to have a small effect. Ill-conditioned only requires some small perturbation to have a large effect. Condition number of a problem Absolute Relative

7 Stability An algorithm is stable if F(x) f(x ) for some x x. This says that a stable algorithm conputes "nearly the right answer" to "nearly the right question". Notice the contrast between conditioning and stability: Conditioning applies to problems. Stability applies to algorithms.

8 Accuracy Stability plus good conditioning implies accuracy. If a stable algorithm is applied to a well-conditioned problem, then F(x) f(x). Conversely, if a problem is ill-conditioned, an accurate solution may not be possible or even meaningful. We cannot ask more of an algorithm than stability.

9 Examples Addition, subtraction, multiplication, division. Addition, multiplication, division with positive numbers are well-conditioned problems. Subtraction is not. Zeros of a quadratic equation The problem of computing the two roots is well-conditioned. However, the quadratic formula is not a stable algorithm. Solving systems of linear equations Ax = b. Conditioning depends on the matrix A.

10 Floating-point Arithmetic The floating-point numbers F are a subset of the real numbers. For a real number x, let fl(x) F denote the floating point approximation to x. Let and represent the four floating point and exact arithmetic operations. Typically, there is a number u << 1 called machine epsilon, such that fl(x) = x(1 + ε) for some ε with ε u. a, b F, a b = (a b)(1 + ε) for some ε with ε u.

11 Stability of Floating Point Arithmetic Floating point arithmetic is stable for computing sums, products, quotients, and differences of two numbers. Sequences of these operations can be unstable however. Example Assume 10 digit precision ( ) - 1 = (1-1) = Floating point operations are not always associative.

12 More Bad Example Calculating e-a with a > 0 by Taylor Series. The round-off error is approximately u times the largest partial sum. Calculating e a and then taking its inverse gives a full-precision answer Roots of a quadratic (ax2 + bx + c) If x 1 0 and x 2 >> 0, then the quadratic formula is unstable. Computing x 2 by the quadratic formula and then setting x 1 = cx 2 / a is stable.

13 Backward Error Analysis Backward error analysis is a method of analyzing roundoff error and assessing stability. We want to show that the result of a floating-point operation has the same effect as if the original data had been perturbed by an amount in O(u). If we can show this, then the algorithm is stable.

14 More examples Matrix factorization Generally ill-conditioned. There are stable algorithms, however. Zeros of a polynomial Generally ill-conditioned. Eigenvalues of a matrix For a symmetric matrix, finding eigenvalues is wellconditioned, finding eigenvectors is ill-conditioned. For non-symmetric matrices, both are ill-conditioned. In all cases, there are stable algorithms.